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Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs

Published in SIAM/ASA Journal on Uncertainty Quantification, 2018

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. Download paper

Recommended citation: D. Torlo, F. Ballarin, and G. Rozza. (2018). "Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs." SIAM/ASA Journal on Uncertainty Quantification, 6(4): 1475--1502. https://doi.org/10.1137/17M1163517

Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification

Published in Journal of Computational and Applied Mathematics, 2019

In this work, we present a model order reduction (MOR) technique for hyperbolic conservation laws with applications in uncertainty quantification (UQ). Download paper here

Recommended citation: R. Crisovan, D. Torlo, R. Abgrall, and S. Tokareva. (2019). "Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification." Journal of Computational and Applied Mathematics, 348:466 – 489. https://doi.org/10.1016/j.cam.2018.09.018

Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases

Published in arXiv, 2020

In this work, we study MOR algorithms for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. Download paper

Recommended citation: D. Torlo. (2020). "Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases." arXiv preprint, arXiv:2003.13735. https://arxiv.org/abs/2003.13735

High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

Published in SIAM Journal on Scientific Computing, 2020

This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. Download paper

Recommended citation: R. Abgrall, and D. Torlo. (2020). "High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models." SIAM Journal on Scientific Computing, 42(3): B816--B845. https://doi.org/10.1137/19M128973X

Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

Published in Applied Numerical Mathematics, 2020

Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Download paper

Recommended citation: P. Öffner and D. Torlo. (2020). "Arbitrary high-order, conservative and positivity preserving Patankar--type deferred correction schemes." Applied Numerical Mathematics, 153:15 – 34. https://doi.org/10.1016/j.apnum.2020.01.025

DeC and ADER: Similarities, Differences and a Unified Framework

Published in Journal of Scientific Computing, 2021

In this paper, we demonstrate that the explicit ADER approach can be seen as a special interpretation of the deferred correction (DeC) method. Download paper

Recommended citation: M. H. Veiga, P. Öffner, and D. Torlo. (2021). "DeC and ADER: Similarities, Differences and a Unified Framework." Journal of Scientific Computing, 87, 2 (2021). https://doi.org/10.1007/s10915-020-01397-5. https://doi.org/10.1007/s10915-020-01397-5

Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

Published in arXiv, 2021

In this paper, we study different high order FEM methods for hyperbolic problems, providing parameters that lead to stable and reliable schemes. Download paper

Recommended citation: Michel, S., Torlo, D., Ricchiuto, M. and Abgrall, R.. Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping. J Sci Comput 89, 31 (2021). https://doi.org/10.1007/s10915-021-01632-7 https://doi.org/10.1007/s10915-021-01632-7

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

Published in arXiv, 2021

In this paper, we study different high order FEM methods for hyperbolic problems, providing parameters that lead to stable and reliable schemes. Download paper

Recommended citation: R. Abgrall, E. Le Mélédo, P. Öffner and D. Torlo. (2021). "Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes. " arXiv preprint, https://arxiv.org/abs/2106.05005. https://arxiv.org/abs/2106.05005

A New Stability Approach for Positivity-preserving Patankar-type Schemes

Published in arXiv, 2021

We study a new type of stability for a class of positivity-preserving nonlinear schemes (Patankar-type schemes) and we discover two types of issues: oscillations around stady states when the timestep is too large and spurious steady states where some methods get stuck. Download paper

Recommended citation: D. Torlo, P. Öffner and H. Ranocha. (2021). "A New Stability Approach for Positivity-preserving Patankar-type Schemes. " arXiv preprint, https://arxiv.org/abs/2108.07347. https://arxiv.org/abs/2108.07347

Analytical traveling vortex solutions of hyperbolic equations for validating very high order schemes

Published in arXiv, 2021

Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like analytical solutions in 2 dimensions for the shallow water equations (and, hence, Euler equations) that can be used to test the order of accuracy of numerical methods. These solutions have different smoothness in their derivatives (up to arbitrary derivatives) and can be used accordingly to the order of accuracy of the scheme to test. Download paper

Recommended citation: M. Ricchiuto and D. Torlo. (2021). "Analytical traveling vortex solutions of hyperbolic equations for validating very high order schemes. " arXiv preprint, https://arxiv.org/abs/2109.10183. https://arxiv.org/abs/2109.10183

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Published in arXiv, 2021

In shallow water equations simulations the positivity of water height is a fundamental property to preserve. We use a linearly implicit modified Patankar Deferred Correction method to guarantee its positivity without any restriction on the time step. The rest of the discretization is obtained with a classical WENO5 finite volume method. Download paper

Recommended citation: M. Ciallella, L. Micalizzi, P. Öffner and D. Torlo. (2021). "An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations. " arXiv preprint, https://arxiv.org/abs/2108.07347. https://arxiv.org/abs/2108.07347

talks

ADER and DeC: Arbitrarily High Order Explicit Time Integration Methods

Published:

When we think about high order time integration, Runge–Kutta (RK) is the most known class of schemes that comes to mind, for historical reasons. Nevertheless, many other techniques have been developed during these years to improve these techniques and to get a generalized form of them.

Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes

Published:

Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

High Order Well-Balanced Discrete Kinetic Model for Shallow Water Equations

Published:

In this work, we study a kinetic model that contains stiff relaxation terms in the source. This model can be applied to any non linear hyperbolic problem without source term, that we will call macroscopic problem, to obtain a larger system of equations with linear fluxes and non–linear source terms, the microscopic problem, that converges asymptotically to the original hyperbolic system.

Continuous Galerkin high order well-balanced discrete kinetic model for shallow water equations

Published:

Kinetic models describe many physical phenomena, inter alia Boltzmann equations, but can also be used to approximate with an artificial relaxation procedure other macroscopic models. We consider the kinetic model proposed by Aregba-Driollet and Natalini \cite{natalini}, and we modify it in order to approximate shallow water (SW) equations. The difference with the original model stands in the presence of the source term in the SW equations due to the effect of the bathymetry. Thus, the kinetic model \cite{natalini} must be extended in order to include this term and to maintain the asymptotic convergence to the macroscopic limit of the SW problem.

teaching